For each \( n \in \mathbb {N} \)  and \( r^h =F^h, S^h \)  and \( G^h \) , define
\(r_n^h(u)= {\left\lbrace \begin{array}{ll}r^h(u) &\text{if } |u|_{\mathbb {E}_h} \le n \\r^h\left(\frac{n u}{|u|_{\mathbb {E}_h}} \right)&\text{if } |u|_{\mathbb {E}_h} > n.\end{array}\right.}\) 

Then \( F_{n}^h \) , \( S_n^h \)  and \( G_n^h \)  are Lipschitz on \( \mathbb {E}_h \) .
Fix \( n \in \mathbb {N} \) .
Let \( A_n: \mathcal {E}_h \rightarrow \mathcal {E}_h \)  be given by
\(\begin{aligned}A_n(u)(t)&= m_0 + \int _0^t \left(F_{n}^h(u(s)) + \frac{1}{2} S_n^h(u(s)) \right)ds + \int _0^t G_n^h (u(s)) \ dW(s) \\&= m_0 + I_n(t) + J_n(t).\end{aligned}\) 
We first verify that \( A_n(u) \in \mathcal {E}_h \)  for \( u \in \mathcal {E}_h \) .
Note that \( F_n^h \)  and \( S_n^h \)  are bounded on \( \mathbb {E}_h \) , with
\(\mathbb {E}\left[ |I_n(t)|_{\mathbb {E}_h}^2 \right]&\le T \mathbb {E}\left[ \int _0^t \left| F_{n}^h(u(s)) + \frac{1}{2} S_n^h(u(s)) \right|_{\mathbb {E}_h}^2 ds \right] \\&\le C_1(h,n) T \mathbb {E}\left[ \int _0^t |u(s)|_{\mathbb {E}_h}^2 ds \right] \\&\le C_1(h,n) T^2 |u|_{\mathcal {E}_h}^2,\) 

for some constant \( C_1 \)  that depends on \( h \)  and \( n \) .
For \( J_n(t) \) , we have
\(\sum _j q_j^2 |f_j G^h(u(s))|_{\mathbb {L}_h^2}^2&= \sum _j q_j^2 \left| f_j u(s) \times \left(u(s) \times \partial ^h u(s) \right)+ \gamma f_j u(s) \times \partial ^h u(s) \right|_{\mathbb {L}_h^2}^2 \\&\le 2 |\kappa ^2|_{\mathbb {L}_h^\infty } \left(|u(s)|_{\mathbb {L}_h^\infty }^4 + \gamma ^2 |u(s)|_{\mathbb {L}_h^\infty }^2 \right)|\partial ^h u(s)|^2_{\mathbb {L}_h^2}.\) 

where the last inequality holds by Tonelli's theorem.
This together with Remark REF (b) implies
\(\sum _j q_j^2 |\partial ^{h} (f_j G^h(u(s))) |_{\mathbb {L}_h^2}^2&\le \frac{4}{h^2} \sum _j q_j^2 |f_j G^h(u(s))|_{\mathbb {L}_h^2}^2 \\&\le \frac{8}{h^2} |\kappa ^2|_{\mathbb {L}_h^\infty } \left(|u(s)|_{\mathbb {L}_h^\infty }^4 + \gamma ^2 |u(s)|_{\mathbb {L}_h^\infty }^2 \right)\ |\partial ^{h} u(s)|^2_{\mathbb {L}_h^2}.\) 

Then by the definition of \( G_n^h \) , the assumption (REF ) and Fubini's theorem,
\(\mathbb {E}\left[ \int _0^t \sum _j q_j^2 \left| f_j G_n^h (u(s)) \right|^2_{\mathbb {H}_h^1} \ ds \right]&\le 2 C_\kappa ^2 (n^4 + \gamma ^2 n^2) \left(1+ \frac{4}{h^2} \right)T \sup _{s \in [0,t]} \mathbb {E}\left[ |u(s)|_{\mathbb {E}_h}^2 \right] \\&= C_2(h,n,\kappa ) T |u|_{\mathcal {E}_h}^2,\) 

for \( C_2(h,n,\kappa ) = 2C_\kappa ^2 (n^4 + \gamma ^2 n^2) (1+\frac{4}{h^2}) T\) .
Thus, \( J_n \)  is a \( \mathbb {H}_h^1 \) -valued continuous martingale.
By Lemma REF  (or Remark REF (c)), there exists a constant \( C>0 \)  such that
\(\begin{aligned}| J_n(t) |_{\mathbb {E}_h}^2&= | J_n(t) |_{\mathbb {L}_h^\infty }^2 + | \partial ^{h} J_n(t) |_{\mathbb {L}_h^2}^2\le (C^2+1) |J_n(t) |_{\mathbb {H}_h^1}^2.\end{aligned}\) 
From [1]},
\(\begin{aligned}\mathbb {E}\left[ \sup _{t \in [0,T]} \left| J_n(t) \right|_{\mathbb {H}_h^1}^2 \right]&= \mathbb {E}\left[ \sup _{t \in [0,T]} \left| \int _0^t G_n^h(u(s)) \ dW(s) \right|_{\mathbb {H}_h^1}^2 \right] \\&\le \mathbb {E}\left[ \int _0^T \sum _j q_j^2 \left| f_j G_n^h(u(s)) \right|_{\mathbb {H}_h^1}^2 \ ds \right] \\&\le C_2(h,n,\kappa ) T |u|_{\mathcal {E}_h}^2.\end{aligned}\) 
It follows from (REF ) and (REF ) that
\(\mathbb {E} \left[ \sup _{t \in [0,T]} \left| J_n(t) \right|_{\mathbb {E}_h}^2 \right]\le \left(C^2+ 1\right)C_2(h,n,\kappa ) T |u|_{\mathcal {E}_h}^2.\) 
Thus, \( A_n(u) \in \mathcal {E}_h \)  for \( u \in \mathcal {E}_h \) .
For \( u, \nu \in \mathcal {E}_h \) , there exists a constant \( C >0 \)  such that
\(|A_n(\nu ) - A_n(u)|_{\mathcal {E}_h}^2&\le C \sup _{t \in [0,T]} \mathbb {E} \left[ \left| \int _0^t F_{n}^h(\nu (s)) - F_{n}^h(u(s)) \ ds \right|_{\mathbb {E}_h}^2 \right] \\&\quad + C \sup _{t \in [0,T]} \mathbb {E} \left[ \left| \int _0^t \frac{1}{2} \left(S_n^h(\nu (s)) - S_n^h(u(s)) \right)\ ds \right|_{\mathbb {E}_h}^2 \right] \\&\quad + C \sup _{t \in [0,T]} \mathbb {E} \left[ \left| \int _0^t \left(G_n^h(\nu (s)) - G_n^h(u(s)) \right)dW(s) \right|_{\mathbb {E}_h}^2 \right].\) 

Similarly, \( M_n(t) := \int _0^t \left(G_n^h(\nu (s)) - G_n^h(u(s)) \right)dW(s) \)  is a \( \mathbb {H}_h^1 \) -valued continuous martingale, and replacing \( J_n \)  by \( M_n \)  in (REF ) and (REF ), we have
\(\mathbb {E} \left[ \sup _{t \in [0,T]} \left| M_n(t) \right|_{\mathbb {E}_h}^2 \right]\le \mathbb {E} \left[ \int _0^T \sum _j q_j^2 \left| f_j \left(G_n^h(\nu (s)) - G_n^h(u(s))\right)\right|_{\mathbb {H}_h^1}^2 \ ds \right].\) 
By construction, if \( |\nu (s)|_{\mathbb {E}_h}, |u(s)|_{\mathbb {E}_h} \le n \) , then
\(|G_n^h(\nu (s)) - G_n^h(u(s)) |_{\mathbb {L}_h^2}& \le |\nu (s) - u(s)|_{\mathbb {L}_h^\infty } \left(|\nu (s) |_{\mathbb {L}_h^\infty } + |u(s)|_{\mathbb {L}_h^\infty } +|\gamma | \right)|\partial ^h \nu (s)|_{\mathbb {L}_h^2} \\&\quad + \left(|u(s)|_{\mathbb {L}_h^\infty }^2 + |\gamma | |u(s)|_{\mathbb {L}_h^\infty } \right)| \partial ^h \nu (s) - \partial ^h u(s) |_{\mathbb {L}_h^2} \\&\le (3 n^2 + 2|\gamma |n) \ |\nu (s) - u(s) |_{\mathbb {E}_h}.\) 

If \( |\nu (s)|_{\mathbb {E}_h}, |u(s)|_{\mathbb {E}_h} > n \) , then let \( n_s^\nu = n |\nu (s)|_{\mathbb {E}_h}^{-1} \)  and \( n_s^u = n |u(s)|_{\mathbb {E}_h}^{-1} \) , we have
\(|n_s^\nu - n_s^u| \ |u|_{\mathbb {E}_h} \le |\nu (s)- u(s)|_{\mathbb {E}_h},\) 

and
\(&|G_n^h(\nu (s)) - G_n^h(u(s)) |_{\mathbb {L}_h^2} \\&= |G^h(n_s^\nu \nu (s)) - G^h(n_s^u u(s)) |_{\mathbb {L}_h^2} \\&\le \left(n_s^\nu |\nu (s) - u(s)|_{\mathbb {L}_h^\infty } + | n_s^\nu - n_s^u| |u(s)|_{\mathbb {L}_h^\infty } \right)\left(n_s^\nu |\nu (s) |_{\mathbb {L}_h^\infty } + n_s^u |u(s)|_{\mathbb {L}_h^\infty } + |\gamma | \right)n_s^\nu |\partial ^h \nu (s)|_{\mathbb {L}_h^2} \\&\quad + \left(n_s^\nu | \partial ^h \nu (s) - \partial ^h u(s) |_{\mathbb {L}_h^2} + | n_s^\nu - n_s^u| |\partial ^h u(s)|_{\mathbb {L}_h^2} \right)\left((n_s^u)^2 |u(s)|_{\mathbb {L}_h^\infty }^2 + |\gamma | n_s^u |u(s)|_{\mathbb {L}_h^\infty } \right)\\&\le 2 (3n^2 + 2|\gamma |n) \ |\nu (s) - u(s) |_{\mathbb {E}_h}.\) 

If \( |\nu (s)|_{\mathbb {E}_h} > n \)  and \( |u(s)|_{\mathbb {E}_h} \le n \) , then
\(|n_s^\nu -1 |\le n^{-1} |n-|\nu (s)|_{\mathbb {E}_h}|\le n^{-1} | |u(s)|_{\mathbb {E}_h} - |\nu (s)|_{\mathbb {E}_h} |\le n^{-1} |u(s)-\nu (s)|_{\mathbb {E}_h},\) 

implying that \( |n_s^\nu - 1| \ |u|_{\mathbb {E}_h} \le |u(s)- \nu (s)|_{\mathbb {E}_h} \)  and
\(|G_n^h(\nu (s)) - G_n^h(u(s)) |_{\mathbb {L}_h^2}&\le 2 (3n^2 + 2|\gamma |n) \ |\nu (s) - u(s) |_{\mathbb {E}_h}.\) 

Similar result follow for \( \partial ^h (G_n^h(\nu )-G_n^h(u)) \)  using Remark REF (b).
Then, by (REF ), Lemma REF  and Hölder's inequality, there exist constants \( L_1(h,n,T) \)  and \( L_2(h,n,T) \)  such that
\(|A_n(\nu ) - A_n(u)|_{\mathcal {E}_h}^2&\le L_1(h,n,T) \sup _{t \in [0,T]} \mathbb {E} \left[ \int _0^t |\nu (s) - u(s)|_{\mathbb {E}_h}^2 \ ds\right] \\&\le L_2(h,n,T) |\nu -u|_{\mathcal {E}_h}^2.\) 

Consider the discrete equation
\(dm^h_n(t) = \left(F_{n}^h(m^h_n(t)) + \frac{1}{2} S_n^h(m^h_n(t)) \right)dt + G_n^h(m^h_n(t)) \ dW(t),\) 
with \( m^h_n(0) = m_0 \in \mathbb {E}_h \)  on intervals \( [(k-1) \tilde{T},k\tilde{T}] \)  for \( k \ge 1 \) , where \( \tilde{T} \)  satisfies \( L_2(h,n,\tilde{T}) < 1 \) .
By the Banach fixed point theorem, there exists a unique solution \( m_n^h \in \mathcal {E}_h \)  of (REF ) on \( [0,T] \) .
Define the stopping times
\(\tau _n := \inf \lbrace  t \ge 0 : |m_n^h(t)|_{\mathbb {E}_h} > n \rbrace , \quad \tau _n^{\prime } := \inf \lbrace  t\ge 0 : |m_{n+1}^h(t)|_{\mathbb {E}_h} > n \rbrace .\) 

Let \( \tau = \tau _n \wedge \tau _n^{\prime } \) .
Then \( A_n(m_{n+1}^h) = A_{n+1}(m_{n+1}^h) \)  on \( [0,\tau ) \) , and by (REF ),
\(\mathbb {E} \left[ \sup _{t \in [0,T]} \left| m_{n+1}^h(t \wedge \tau ) - m_n^h(t \wedge \tau ) \right|_{\mathbb {E}_h}^2 \right]&= \mathbb {E} \left[ \sup _{t \in [0,T]} \left| A_n(m_{n+1}^h)(t \wedge \tau ) - A_n(m_n^h)(t \wedge \tau ) \right|_{\mathbb {E}_h}^2 \right] \\&\le L_1(h,n,T) \mathbb {E} \left[ \int _0^T \left| m_{n+1}^h(s) - m_n^h(s) \right|_{\mathbb {E}_h}^2 \ ds \right],\) 

which implies
\(\mathbb {E} \left[ \sup _{t \in [0,T]} \left| m_{n+1}^h(t \wedge \tau ) - m_n^h(t \wedge \tau ) \right|_{\mathbb {E}_h}^2 \right] = 0,\) 

by Grönwall's lemma.
Thus, \( m_{n+1}^h(\cdot \wedge \tau ) = m_n^h(\cdot \wedge \tau ) \)  and \( \tau = \tau _n \) , \( \mathbb {P} \) -a.s. and the discrete equation (REF ) admits a local solution \( m^h(t) = m_n^h(t) \)  for \( t \in [0,\tau _n] \) .
Recall that \( m_0 \in \mathbb {E}_h \)  and \( |m_0(x)| =1 \)  for all \( x \in \mathbb {Z}_h \) .
Applying Itô's lemma to \( \frac{1}{2}|m^h(t,x)|^2 \) ,
\(\frac{1}{2} d|m^h(t,x)|^2&= \left\langle F^h(m^h(t))(x) + \frac{1}{2} S^h(m^h(t))(x), m^h(t,x) \right\rangle dt + \frac{1}{2} \kappa ^2(x) |G^h(m^h(t))(x) |^2 dt \\&\quad + \left\langle G^h(m^h(t))(x), m^h(t,x) \right\rangle dW(t).\) 

By \( \langle a, a \times b \rangle =0 \) , for any \( t \in [0,\tau _n] \)  and \( x \in \mathbb {Z}_h \) ,
\(&\left\langle F^h(m^h), m^h \right\rangle (t,x) = 0, \\&\left\langle S^h(m^h), m^h \right\rangle (t,x)= \left\langle \kappa ^2 G^h_2(m^h), m^h \right\rangle (t,x)= - \kappa ^2(x) |G^h(m^h)|^2(t,x), \\&\left\langle G^h(m^h), m^h \right\rangle (t,x) = 0.\) 

Therefore, \( |m^h(t,x)| = |m_n^h(t,x)| = |m_0(x)| = 1 \)  for any \( t \in [0,\tau _n] \)  and \( x \in \mathbb {Z}_h \) .
For any fixed \( h \)  and \( n \) , the unique solution \( m_n^h \)  of (REF ) satisfies
\(\mathbb {E} \left[ |m^h_n(t \wedge \tau _n)|^2_{\mathbb {E}_h} \right]&= \mathbb {E} \left[ |m^h_n(t \wedge \tau _n)|_{\mathbb {L}_h^\infty }^2 + |\partial ^{h} m^h_n(t \wedge \tau _n) |_{\mathbb {L}_h^2}^2 \right] \\&= 1 + \mathbb {E} \left[ |\partial ^{h} m^h_n(t \wedge \tau _n) |_{\mathbb {L}_h^2}^2 \right].\) 

We apply Itô's lemma to \( \frac{1}{2}|\partial ^{h} m_n^h(t \wedge \tau _n)|^2_{\mathbb {L}_h^2} \) ,
\(&\frac{1}{2}|\partial ^{h} m_n^h(t \wedge \tau _n)|^2_{\mathbb {L}_h^2}- \frac{1}{2}|\partial ^{h} m_0|^2_{\mathbb {L}_h^2} \\&= \int _0^{t \wedge \tau _n} \left\langle -\Delta ^h m_n^h(s), F^h(m_n^h(s)) + \frac{1}{2} S^h(m_n^h(s)) \right\rangle _{\mathbb {L}_h^2} ds \\&\quad + \int _0^{t \wedge \tau _n} \frac{1}{2} \sum _j q_j^2 \left| \partial ^{h}\left( f_j G^h(m_n^h(s))\right) \right|_{\mathbb {L}_h^2}^2 ds+ \int _0^{t \wedge \tau _n} \left\langle -\Delta ^h m_n^h(s), G^h(m_n^h(s)) \ dW(s) \right\rangle _{\mathbb {L}_h^2} .\) 

Since \( |m_n^h(t,x)|=1 \)  for \( (t,x) \in [0,\tau _n] \times \mathbb {Z}_h \) , and
\(|\Delta ^h m_n^h(t)|_{\mathbb {L}_h^2}^2 + |\partial ^{h} m_n^h(t)|_{\mathbb {L}_h^2}^4 \le \frac{8}{h^2} |\partial ^{h} m_n^h(t)|_{\mathbb {L}_h^2}^2,\) 

there exist constants \( \beta _1 \)  and \( \beta _2 \)  that depend on \( h \)  (not \( n \) ) such that
\(\left\langle -\Delta ^h m_n^h(s), F^h(m_n^h(s)) + \frac{1}{2} S^h(m_n^h(s)) \right\rangle _{\mathbb {L}_h^2}+ \frac{1}{2} \sum _j q_j^2 \left|\partial ^{h}\left(f_j G^h(m_n^h(s)) \right)\right|_{\mathbb {L}_h^2}^2\le \beta _1(h) |\partial ^{h} m_n^h(s)|_{\mathbb {L}_h^2}^2,\) 

and
\(\left\langle \Delta ^h m_n^h(s), G^h(m_n^h(s)) \right\rangle _{\mathbb {L}_h^2}^2&\le \frac{1}{4}\left(|m_n^h(s) \times \Delta ^h m_n^h(s)|^2_{\mathbb {L}_h^2} + |\partial ^h m_n^h(s)|^2_{\mathbb {L}_h^2} \right)^2 \\&\le \beta _2(h) |\partial ^{h} m_n^h(s)|_{\mathbb {L}_h^2}^2.\) 

Then, by the boundedness of \( \kappa ^2 \) , the stochastic integral \( \int _0^{t \wedge \tau _n} \langle \Delta ^h m_n^h(s), G^h(m_n^h(s))\ dW(s) \rangle _{\mathbb {L}_h^2} \)  is a square integrable continuous martingale for \( m_n^h \in \mathcal {E}_h \) , for every \( h>0 \) .
Now, we have
\(|\partial ^{h} m_n^h(t \wedge \tau _n)|^2_{\mathbb {L}_h^2}&\le |\partial ^{h} m_0|^2_{\mathbb {L}_h^2} + 2\int _0^{t \wedge \tau _n} \beta _1(h) |\partial ^{h} m_n^h(s)|^2_{\mathbb {L}_h^2} \ ds \\&\quad - 2\int _0^{t \wedge \tau _n} \left\langle \Delta ^h m_n^h(s), G^h(m^h_n)(s) \ dW(s) \right\rangle _{\mathbb {L}_h^2} ,\) 

where the stochastic integral part vanishes after taking expectation.
We obtain
\(\mathbb {E} \left[ |m^h_n(t \wedge \tau _n)|^2_{\mathbb {E}_h} \right]&\le \mathbb {E} \left[ |m_0|^2_{\mathbb {E}_h} + 2\beta _1(h) \int _0^{t } |m^h_n(s\wedge \tau _n)|^2_{\mathbb {E}_h} \ ds \right].\) 
 By Grönwall's lemma,
\(\mathbb {E} \left[ |m^h_n(t \wedge \tau _n)|^2_{\mathbb {E}_h} \right] \le \mathbb {E} \left[ |m_0|^2_{\mathbb {E}_h} \right] \exp \left(\int _0^{t} 2\beta _1(h) ds \right)\le K(h,t).\) 
By the definition of \( \tau _n \) , the left-hand-side of (REF ) is greater than \( n^2 \mathbb {P}(\tau _n \in [0,t]) \) , thus
\(\lim _{n \rightarrow \infty } \mathbb {P}(\tau _n \in [0,t]) \le \lim _{n \rightarrow \infty } K(h,t) n^{-2} = 0.\) 

In other words, \( \tau _n \rightarrow \infty \) , \( \mathbb {P} \) -a.s, as \( n \rightarrow \infty \) .
Thus, the process
\( m^h(t) = \lim _{n \rightarrow \infty } m_n^h(t \wedge \tau _n) \)  is the unique solution of the semi-discrete scheme (REF ).
